residualvektor

Residualvektor is a term that arises in the context of solving systems of linear equations or approximations. When an iterative method is used to find a solution to an equation, a residualvektor represents the difference between the exact solution and the current approximation. Specifically, if $Ax = b$ is the system of equations and $x_k$ is the approximation at iteration $k$, then the residualvektor $r_k$ is defined as $r_k = b - Ax_k$. This vector quantifies how far the current approximation $x_k$ is from satisfying the original equation.

The properties and magnitude of the residualvektor are crucial for assessing the convergence and accuracy of iterative algorithms. A residualvektor close to the zero vector indicates that the current approximation is a good solution to the system. Many iterative solvers aim to minimize the norm of the residualvektor. In numerical analysis, the residualvektor is often used to estimate the error in the solution. However, it is important to note that a small residualvektor does not always guarantee a small error in the solution, especially for ill-conditioned systems. The concept of residualvektor is fundamental in fields such as numerical linear algebra and optimization.